Related papers: A bound on the equivariant unknotting number

We define an invariant of transverse links in the standard contact 3-sphere as a distinguished element of the Khovanov homology of the link. The quantum grading of this invariant is the self-linking number of the link. For knots, this gives…

The fundamental problem of knot theory is to know whether two knots are equivalent or not. As a tool to prove that two knots are different, mathematicians have developed various invariants. Knots invariants are just functions that can be…

We introduce a Poincar\'{e} polynomial with two-variable $t$ and $x$ for knots, derived from Khovanov homology, where the specialization $(t, x)$ $=$ $(1, -1)$ is a Vassiliev invariant of order $n$. Since for every $n$, there exist…

Ito-Takimura recently defined a splice-unknotting number $u^-(D)$ for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We…

In this work we describe a new invariant of virtual knots. We show that this transcendental function invariant generalizes several polynomial invariants of virtual knots, such as the writhe polynomial, the affine index polynomial and the…

In this note we use Blanchfield forms to study knots that can be turned into an unknot using a single $\overline{t}_{2k}$ move.

This paper, to be regularly updated, lists those prime knots with the fewest possible number of crossings for which values of basic knot invariants, such as the unknotting number or the smooth 4-genus, are unknown. This list is being…

We show that the forbidden detour move, essentially introduced by Kanenobu and Nelson, is an unknotting operation for virtual knots. Then we define the forbidden detour number of a virtual knot to be the minimal number of forbidden detour…

This paper generalizes the bordered-algebraic knot invariant introduced in an earlier paper, giving an invariant now with more algebraic structure. It also introduces signs to define these invariants with integral coefficients. We describe…

Vassiliev introduced filtered invariants of knots using an unknotting operation, called crossing changes. Goussarov, Polyak, and Viro introduced other filtered invariants of virtual knots, which order is called GPV-order, using an…

Let $\nu$ be either the Ozsv\'ath-Szab\'o $\tau$-invariant or the Rasmussen $s$-invariant, suitably normalized. For a knot $K$, Livingston and Naik defined the invariant $t_\nu(K)$ to be the minimum of $k$ for which $\nu$ of the $k$-twisted…

Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild…

In this paper we introduce the notion of an unknotting index for virtual knots. We give some examples of computation by using writhe invariants, and discuss a relationship between the unknotting index and the virtual knot module. In…

We develop a reinforcement learning pipeline for simplifying knot diagrams. A trained agent learns move proposals and a value heuristic for navigating Reidemeister moves. The pipeline applies to arbitrary knots and links; we test it on…

In this paper, we show that if a diagram of a surface-knot $F$ has at most three triple points, then the cocyle invariant of $F$ is an integer. In particular, for a surface-knot of genus one, the triple point number invariant is at least…

Yang-Baxter operators (YBOs) have been employed to construct quantum knot invariants. More recently, cohomology theories for YBOs have been independently developed, drawing inspiration from analogous theories for quandles and other discrete…

We will strengthen the known upper and lower bounds on the delta-crossing number of knots in therms of the triple-crossing number. The latter bound turns out to be strong enough to obtain (unknown values of) triple-crossing numbers for a…

We introduce an unknotting-type number of knot projections that gives an upper bound of the crosscap number of knots. We determine the set of knot projections with the unknotting-type number at most two, and this result implies classical…

We relate certain abelian invariants of a knot, namely the Alexander polynomial, the Blanchfield form, and the Arf invariant, to intersection data of a Whitney tower in the 4-ball bounded by the knot. We also give a new 3-dimensional…

We consider the probability of knotting in equilateral random polygons in Euclidean 3-dimensional space, which model, for instance, random polymers. Results from an extensive Monte Carlo dataset of random polygons indicate a universal…